Matching with Contracts

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Matching with Contracts

Fuhito Kojima

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Generalized Matching Theory

Overview

The model of two-sided matching proved useful in labor markets and school choice (and possibly other applications).

But the mode was restrictive in several ways:

1 _{Terms of contracts were exogenous: no wage, no working hours, etc.}

2 Colleges (hospitals, schools) have very simple preferences: rank

students linearly. No issue of interaction between students: racial balance and score distribution etc in schools.

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Crawford and Knoer (1981) and Kelso and Crawford (1982) introduce wages in matching. Roth and Sotomayor (1990), Chapter 6 also talks about the issue.

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Matching with Contracts

D andH are sets of doctors and hospitals.

X is a set of contracts. Each contractx ∈X is associated with one doctor

xD ∈D and one hospitalxH ∈H.

Each doctor can sign one contract (or no contract,∅), but a hospital can

sign multiple contracts.

Pi is a strict preference relation fori ∈D∪H.

Ci(X0) is the chosen set of contracts by i fromX0 ⊆X (we write

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Matching with contracts will subsume

1 _{The matching model without contracts: a contract simply specifies}

who is matched with whom.

2 Matching with wages (Kelso and Crawford 1982): a contract specifies

who is matched with whom, at what wage.

Other terms of contracts can be incorporated, such as

1 How long to work.

2 _{What kind of job (research position or clinical position?).}

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The Cadet-Branch Matching in the Military

A recent application for personnel allocation in the military (Sonmez and Switzer 2013, Sonmez 2013).

1 Cadets and branches in the military are to be matched, after

graduating from US military academy (USMA).

2 Branches have priority ordering, called order-of-merit list, based on

academic performance, physical fitness test score, and so on.

3 _{Cadets can commit to work for a longer year (8 instead of the}

standard 5).

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Stability

An allocation X0⊆X is stableif

1 Individual rationality: C_D(X0) =C_H(X0) =X0,and

2 No blocking coalition: there exists no hospitalh and set of contracts

X006=Ch(X0) such that X00=Ch(X0∪X00)⊆CD(X0∪X00).

A stable allocation X0 is called the doctor-optimal (doctor-pessimal)

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Generalized Matching Theory Substitutable Preferences

Substitutes

Contracts are substitutes(or, preferences aresubstitutable) forh if there

are no contractsx,z ∈X and a set of contractsY ⊆X such that

z ∈/ Ch(Y ∪ {z}) andz ∈Ch(Y ∪ {x,z}).

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Substitutable preferences include

1 One-to-one matching.

2 Many-to-one matching with “responsive” preferences, i.e., each

hospital has a fixed quota and a ranking over students, and want to choose best-ranked students up to its quota.

3 Some more complex preferences: for example,

Pc :{s1},{s2,s3},{s2},{s3}.

Such a preference may naturally occur in some labor markets.

4 _{Certain complementarity is excluded: for example,}

Pc :{s1,s2},

i.e., preferences such that {{s1,s2},is the only acceptable option,

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Substitutable preferences allow for

1 Racial balance (sometimes called “racial tie-breaker”).

2 Score distribution (as in NYC school choice).

3 More generally, some affirmative action constraints.

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As before, we want to know whether there is always a stable allocation and, if so, want to know how to find it.

Theorem (Roth and Sotomayor 1990; Hatfield and Milgrom 2005)

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Deferred acceptance algorithm generalized

To prove the existence theorem, we use the cumulative offer process (Ausubel and Milgrom 2002, Hatfield and Milgrom 2005, Hatfield and Kojima 2010). This is a generalization of Gale and Shapley’s deferred acceptance algorithm.

LetA_h(t) be the set of “available contracts forhat stept.” A_h(0) =∅for allh.

Step 1: One (arbitrarily chosen) doctor offers her first choice contract

x1. The hospital that is offered the contract,h1= (x1)H, holds the contract if it is acceptable and rejects it otherwise. Let

Ah1(1) ={x1}, andAh(1) =∅for allh6=h1.

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The algorithm terminates when every unmatched doctor has had every acceptable contract rejected. The algorithm terminates in some finite

numberT of steps.

At that point, the algorithm produces X0=S

h∈HCh(Ah(T)), i.e., the set

of contracts that are held by some hospital at the terminal step T.

Idea: hospital h has accumulated offers inAh(t) by time t, andh always

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Proof of Existence Result

Step 1: Feasibility

We will show: for any h∈H,z with zH =h and t≥2, ifz ∈Ah(t−1)

and zD ∈/ [Ch(Ah(t−1))]D then z ∈/Ch(Ah(t)).

Obvious if no contract is offered, or zD offers a contract, to h at step t.

Thus suppose that a contract xt is offered toh, with (xt)D 6=zD. Let

Y =Ah(t−1)\ {y ∈X|yD ∈ {(xt)D,zD}}. By definition, (xt)D ∈/ YD

and zD ∈/ YD. Since (xt)D is making an offer at step t,

(xt)D ∈/ [Ch(Ah(t−1))]D. Also, by assumptionzD ∈/[Ch(Ah(t−1))]D.

Therefore z ∈/ Ch(Ah(t−1)) =Ch(Y ∪z). By bilateral substitutes,

z ∈/ Ch(Y ∪ {z} ∪ {xt}) and hence z ∈/ Ch(Ah(t)).1

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Step 2: Stability

Let X0 be the allocation generated by the algorithm.

(1) Individual rationality: (i) at each step, doctors make offers that are acceptable to them, and (ii) hospitals always choose the best contracts from available contracts.

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In matching without contracts, substitutable preferences are necessary to guarantee that there is a stable matching (Sonmez and Unver, 2010, Hatfield and Kojima 2008).

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Bilateral Substitutes

Contracts arebilateral substitutes(Hatfield and Kojima, 2010) forh

if there are no contracts x,z ∈X and a set of contracts Y ⊆X such that z ∈/Ch(Y ∪ {z}),z ∈Ch(Y ∪ {x,z}), and xD,zD ∈/ YD.

Bilateral substitutes is a weaker condition than substitutes in two ways.

First, when we consider a rejected contract z, we only consider sets of

other contracts that do not involvezD. Second, when we consider a

contract x that may be added to the set of contracts, we only consider

contracts with doctors not in YD.

In a matching problem without contracts (i.e., for any two contracts

x,x0∈X,xD =xD0 andxH =xH0 implyx =x0), the bilateral substitutes

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Application: Stable Matching with Couples

We apply our results to matching with couples:

H andC are the sets of hospitals and couples of doctors. A couple

c = (mc,fc) has two members mc andfc. Couples have preferences over

pairs of hospitals (and being unemployed). Each hospital has one position.

Matching with couples can be seen as a special instance of matching with contracts as follows. Each couple can sign at most two contracts (one for each member), and each hospital can sign one contract (note that a couple in a couple problem plays the role of a hospital in our contract setting, and a hospital in a couple problem plays the role of a doctor). For any couple c = (mc,fc) and hospitalh, there are two possible contracts

between c andh: one of them is a contract that prescribes “to match mc

of couple c to hospital h” and the other is a contract “to matchfc to h.

With this interpretation, matching with couples is a special case of matching with contracts. So Theorem 1 implies that bilateral substitutes (of couples’ preferences) is sufficient for the existence of a stable matching with couples.

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Example

There are two hospitals h andh0, and preference relationPc of couple

c = (mc,fc) is:

Pc :{(h0,mc),(h,fc)} c {(h,mc)} c {(h0,mc)} c {(h,fc)},

(here (˜h,ic) means “let hospital ˜h hire individualic”)

The substitutes condition is violated: (h0,mc)∈/ Cc({(h0,mc),(h,mc)}), but

(h0,mc)∈Cc({(h0,mc),(h,mc),(h,fc)}).

But the preferences satisfy the bilateral substitutes condition.

Interpretation of Pc:

h: position with high wage and long working hours,

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Example 2

Klaus, Klijn and Kakamura (2007) show the existence of a stable matching when preferences of couples are “weakly responsiveness” (that is, roughly speaking, couples behave as if two independent individuals).

But some preferences violate (weak) responsiveness and still allow for a stable allocation.

Consider preferencesP_c given by

Pc :{(h,mc)} c {(h0,fc)}.

These preferences violate the (weak) responsiveness condition: if (weakly) responsive, the couple should prefer{(h,mc),(h0,fc)}to{(h,mc)}and

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Unilateral Substitutes

Many key results in matching theory do not hold even if contracts are bilateral substitutes. Thus we consider a strengthening of the bilateral substitutes condition.

Contracts are unilateral substitutes (Hatfield and Kojima 2010) for

h if there are no contractsx,z ∈X and a set of contracts Y ⊆X such that z ∈/Ch(Y ∪ {z}),z ∈Ch(Y ∪ {x,z}) and zD ∈/YD.

Clearly, the substitutes condition implies the unilateral substitutes

condition, and that the unilateral substitutes condition implies the bilateral substitutes condition. All of these conditions coincide in matching

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Doctor-Optimal Stable Allocations

Theorem

Suppose that contracts are unilateral substitutes for every hospital. Then there exists a doctor-optimal stable allocation. The allocation that is produced by the doctor-proposing deferred acceptance algorithm is the doctor-optimal stable allocation.

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Rural Hospital Theorem

The following condition is satisfied in many environments.

Definition (Hatfield and Milgrom 2005) The preferences of hospital

h ∈H satisfy the law of aggregate demandif for all X0 ⊆X00⊆X,

|Ch(X0)| ≤ |Ch(X00)|.

With this condition and unilateral substitutes, we obtain a version of the so-called “rural hospital theorem” (Roth 1986).

Theorem

If hospital preferences satisfy unilateral substitutes and the law of

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Incentives

A mechanism isstrategy-proof if no individual doctor can misreport

preferences and make herself strictly better off.

A mechanism isgroup strategy-proof if no group of doctors can jointly

misreport preferences and make each member strictly better off.

The doctor-optimal stable mechanism is a mechanism which, for any

reported preference profile P, produces the doctor-optimal stable

allocation underP.

Theorem

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Cadet-Branch Matching again

The theory has become general, but is it all that useful?

Sonmez and Switzer (2013) observe that substitutability fails in the cadet-branch matching model.

But they show that unilateral substitutes condition (and the law of aggregate demand) holds.

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Application to Matching with Distributional Constraints

Another, perhaps unexpected, application of matching with contracts:

matching with distributional constraints (Kamada and Kojima, 2013).

They consider design of matching mechanisms when there are upper limit on the number of doctors who can be matched to subsets of hospitals.

1 Geographical distribution (Japanese medical match) 2 Medical specialty balance.

3 Limit on “academic” master program enrollment (China) 4 Limit on state-financed admission to colleges (Hungary, Ukraine)

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Conclusion about Generalized Matching Theory

Matching theory can be extended to include terms of contracts and complex preferences.

A concept of “substitutes” is important for existence of a stable matching, and many other nice properties.

The substitutes condition has been relaxed: (1) Bilateral substitutes is sufficient for the existence of a stable allocation, but few other results hold. (2) Unilateral substitutes restores many results.